Rediscovering Hyperelasticity by Deep Symbolic Regression

TL;DR

This work introduces deep symbolic regression (DSR) to discover interpretable strain-energy functions for rubber-like hyperelastic materials directly from classical datasets, without human-biased model selection. By encoding the continuum mechanics framework in terms of invariants $I_{oldsymbol{C}}$ and $II_{oldsymbol{C}}$ (and optionally Valanis-Landel decomposition), the method yields low-parameter, physically interpretable expressions that accurately predict multi-axial responses across uniaxial, shear, equibiaxial, and biaxial tests. The study demonstrates both invariant-based and stretch-based formulations, shows robustness to measurement noise, and reveals multiple viable energy functions that fit the same data, highlighting the importance of using diverse loading data. The results point to the practical potential of data-driven, bias-free constitutive law discovery for industrial hyperelastic modeling, with opportunities to further constrain searches via domain priors and physics-guided priors.

Abstract

The accurate modeling of the mechanical behavior of rubber-like materials under multi-axial loading constitutes a long-standing challenge in hyperelastic material modeling. This work employs deep symbolic regression as an interpretable machine learning approach to discover novel strain energy functions directly from experimental results, with a specific focus on the classical Treloar and Kawabata data sets for vulcanized rubber. The proposed approach circumvents traditional human model selection biases by exploring possible functional forms of strain energy functions, expressed in terms of both the first and second principal invariants of the right Cauchy-Green tensor. The resulting models exhibit high predictive accuracy for various deformation modes, including uniaxial tension, pure shear, equal biaxial tension, and biaxial loading. This work underscores the potential of deep symbolic regression in advancing hyperelastic material modeling and highlights the importance of considering both invariants in capturing the complex behaviors of rubber-like materials.

Figures (10)

• Figure 1: Expression sampling process of the deep symbolic regression framework applied to derive the strain energy function $\Psi(\mathrm{I}_{\boldsymbol{\mathrm{C}}}, \mathrm{II}_{\boldsymbol{\mathrm{C}}}) = \mathrm{I}_{\boldsymbol{\mathrm{C}}} \,+\, 0.5\ln(\mathrm{II}_{\boldsymbol{\mathrm{C}}})$. The RNN samples tokens given by operations, functions, constants inputs to build an expression tree starting from a root token. The rewards are computed using the NRMSE to train the RNN using a risk-seeking policy gradient.
• Figure 2: Best fit for the Treloar data set of novel model discovered through DSO. The stress-strain responses for UT, PS and EBT are generated from the strain energy function given in \ref{['equ:TreloarBestFitDSO']}.
• Figure 3: Visualization of the contributions of each term $\Psi_{i}$ for $i=1,\dots,4$ in the strain energy \ref{['equ:TreloarBestFitDSO']}. The responses are shown for UT, PS and EBT for the strain range from $0%$ to $700%$.
• Figure 4: Visualization of the contributions of each term $\Psi_{i}$ for $i=1,\dots,4$ in the strain energy \ref{['equ:TreloarBestFitDSO']}. The responses are shown for UT, PS and EBT for the strain range from $0%$ to $300%$.
• Figure 5: Best fit for the Treloar data set using (\ref{['fig:Treloar_Comparison_Model_1']}) the extended tube model ($R^2 = 96.56108483739724%$), (\ref{['fig:Treloar_Comparison_Model_2']}) the non-hyperelastic Shariff model ($R^2 = 96.37595636432934%$) and (\ref{['fig:Treloar_Comparison_Model_3']}) the stretch-based Ogden model ($R^2 = 95.58273523135516%$) for UT, PS and EBT. The used material parameters are listed in \ref{['tab:ExtendedTubeTreloarMaterialParam']}, \ref{['tab:ShariffTreloarMaterialParam']} and \ref{['tab:OgdenTreloarMaterialParam']}, respectively.